A rational difference equation is a nonlinear difference equation of the form where the initial conditions are such that the denominator never vanishes for any . First-order rational difference equation A first-order rational difference equation is a nonlinear difference equation of the form When and the initial condition are real numbers, this difference equation is called a Riccati difference equation. Such an equation can be solved by writing as a nonlinear transformation of another variable which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in . Equations of this form arise from the infinite resistor ladder problem. Solving a first-order equation First approach One approach to developing the transformed variable , when , is to write where and and where . Further writing can be shown to yield Second approach This approach gives a first-order difference equation for instead of a second-order one, for the case in which is non-negative. Write implying , where is given by and where . Then it can be shown that evolves according to Third approach The equation can also be solved by treating it as a special case of the more general matrix equation where all of A, B, C, E, and X are n × n matrices (in this case n = 1); the solution of this is where Application It was shown in that a dynamic matrix Riccati equation of the form which can arise in some discrete-time optimal control problems, can be solved using the second approach above if the matrix C has only one more row than column. References Further reading Simons, Stuart, "A non-linear difference equation," Mathematical Gazette 93, November 2009, 500–504. Algebra Recurrence relations Dynamical systems